Geodesic
Baire and automata
Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2.

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In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science. 
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Simonnet, Pierre; Cagnard, Benoit. Baire and automata. Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2. doi : 10.46298/dmtcs.392. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.392/

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